You can try to trade before they become FA, but you have to find a trade partner to take on the current contract and risks of resigning that player or just lose the picks for a one year rental. You can accumulate picks, but it takes time for most players to develop if they develop at all and if you keep trading away players or losing them, you're just in a continual rebuild mode. At some point, to remain competitive or become a contender you are going to have to pay players, whether it's your own, FAs or both.
Having all that cap space or draft picks don't mean squat once the season starts and you try to win games. The market is whatever others are willing to pay and not just one persons perceived worth. A bottle of water may cost a buck at the local Stop-N-Shop, but in the middle of the desert a rusty tin cup of muddy water will cost you considerably more. You could always refuse and hope to find an oasis over that next sand dune.